Why is it that the mean deviation is always the least when calculated about the median?
In other words, if $x_1,x_2,x_3,\cdots,x_n$ are real numbers and $M$ is the median of these numbers, then $$f(\alpha)=\frac{\sum_{i=1}^n |x_i-\alpha|}{n} $$ is minimum when $\alpha=M$. Why is it so?
Today I chose to edit a post that I found in the "First questions" queue. It began with the words "Hey guys", and I chose to leave them since they expressed the OP's personality and style, even if I don't really like them. To my surprise, saving my edit removed these words from the post. I tried ...
Let $A,B\in M_3(\mathbb{C})$ be invertible matrices such that $AB=BA=X,\,\,A^{T}+A=B^{T}+B=X^{T}+X,$ then
(A) $A=B$
(B) $\det(A-I)=0$
(C) $\det(B-I)=0$
(D) $\det(X-I)=0$
My working:
$AB+(AB)^T=X+X^T\implies AB+B^TA^T=B+B^T\implies (A-I)B+B^T(A^T-I)=O_3$
$\implies (A-I)B+((A-I)B)^T=O_3\implies (A-...
In linear algebra, two matrices
A
{\displaystyle A}
and
B
{\displaystyle B}
are said to commute if
A
B
=
B
A
{\displaystyle AB=BA}
, or equivalently if their commutator
[
A
,
B
]
=
A
B
−
B
A
{\displaystyle [A,B]=AB-BA}
is zero. A set of matrices
A
1...
In https://arxiv.org/pdf/1511.03198.pdf on page 2 it states that the transport map $T$ between two abs. continuous measures $\mu, \nu\in \mathcal P(\mathbb R)$ with strictly positive densities is unique and given by
$$T(x):=\min \{t \in\mathbb R \, \vert \, F_{\nu}(t) \geq F_{\mu}(x) \}$$ and the...
Why is it that the mean deviation is always the least when calculated about the median?
In other words, if $x_1,x_2,x_3,\cdots,x_n$ are real numbers and $M$ is the median of these numbers, then $$f(\alpha)=\frac{\sum_{i=1}^n |x_i-\alpha|}{n} $$ is minimum when $\alpha=M$. Why is it so?
